Geodesic
$\mathbb{A}^1$-connected components of classifying spaces and purity for torsors
Documenta mathematica, Tome 27 (2022), pp. 2657-2689.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

In this paper, we study the Nisnevich sheafification $\mathcal{H}^1_{\text{ét}}(G)$ of the presheaf associating to a smooth scheme the set of isomorphism classes of $G$-torsors, for a reductive group $G$. We show that if $G$-torsors on affine lines are extended, then $\mathcal{H}^1_{\text{ét}}(G)$ is homotopy invariant and show that the sheaf is unramified if and only if Nisnevich-local purity holds for $G$-torsors. We also identify the sheaf $\mathcal{H}^1_{\text{ét}}(G)$ with the sheaf of $\mathbb{A}^1$-connected components of the classifying space $\mathrm{B}_{\text{ét}}G$. This establishes the homotopy invariance of the sheaves of components as conjectured by Morel. It moreover provides a computation of the sheaf of $\mathbb{A}^1$-connected components in terms of unramified $G$-torsors over function fields whenever Nisnevich-local purity holds for $G$-torsors.
Classification : 14F42, 14L15, 19E15
Keywords: torsors, classifying spaces, motivic homotopy theory 
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     author = {Elmanto, Elden and Kulkarni, Girish and Wendt, Matthias},
     title = {\(\mathbb{A}^1\)-connected components of classifying spaces and purity for torsors},
     journal = {Documenta mathematica},
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Elmanto, Elden; Kulkarni, Girish; Wendt, Matthias. \(\mathbb{A}^1\)-connected components of classifying spaces and purity for torsors. Documenta mathematica, Tome 27 (2022), pp. 2657-2689. http://geodesic.mathdoc.fr/item/DOCMA_2022__27__a1/